Problem: Simplify the following expression: $a = \dfrac{36q^3 - 63q^2}{-9q^3}$ You can assume $q \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $36q^3 - 63q^2 = (2\cdot2\cdot3\cdot3 \cdot q \cdot q \cdot q) - (3\cdot3\cdot7 \cdot q \cdot q)$ The denominator can be factored: $-9q^3 = - (3\cdot3 \cdot q \cdot q \cdot q)$ The greatest common factor of all the terms is $9q^2$ Factoring out $9q^2$ gives us: $a = \dfrac{(9q^2)(4q - 7)}{(9q^2)(-q)}$ Dividing both the numerator and denominator by $9q^2$ gives: $a = \dfrac{4q - 7}{-q}$